Some stable difference approximations to a fourth-order parabolic partial differential equation

Authors:
G. Fairweather and A. R. Gourlay

Journal:
Math. Comp. **21** (1967), 1-11

MSC:
Primary 65.68

DOI:
https://doi.org/10.1090/S0025-5718-1967-0221785-2

Corrigendum:
Math. Comp. **22** (1968), 246-247.

MathSciNet review:
0221785

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References | Similar Articles | Additional Information

**[1]**Garrett Birkhoff, Richard S. Varga, and David Young,*Alternating direction implicit methods*, Advances in Computers, Vol. 3, Academic Press, New York, 1962, pp. 189–273. MR**0168131****[2]**J. Crank and P. Nicolson,*A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type*, Proc. Cambridge Philos. Soc.**43**(1947), 50–67. MR**0019410****[3]**Jim Douglas Jr.,*The solution of the diffusion equation by a high order correct difference equation*, J. Math. and Phys.**35**(1956), 145–151. MR**0090875**, https://doi.org/10.1002/sapm1956351145**[4]**E. C. Du Fort & S. P. Frankel, "Stability conditions in the numerical treatment of parabolic differential equations,"*MTAC*, v. 7, 1953, pp. 135-152. MR**15**, 474.**[5]**D. J. Evans,*A stable explicit method for the finite-difference solution of a fourth-order parabolic partial differential equation*, Comput. J.**8**(1965), 280–287. MR**0189264**, https://doi.org/10.1093/comjnl/8.3.280**[6]**Heinz-Otto Kreiss,*Über implizite Differenzmethoden für partielle Differentialgleichungen*, Numer. Math.**5**(1963), 24–47 (German). MR**0155442**, https://doi.org/10.1007/BF01385876**[7]**Milton Lees,*Alternating direction and semi-explicit difference methods for parabolic partial differential equations*, Numer. Math.**3**(1961), 398–412. MR**0142212**, https://doi.org/10.1007/BF01386038**[8]**A. R. Mitchell and G. Fairweather,*Improved forms of the alternating direction methods of Douglas, Peaceman, and Rachford for solving parabolic and elliptic equations*, Numer. Math.**6**(1964), 285–292. MR**0174184**, https://doi.org/10.1007/BF01386076**[9]**B. Numerov,*Publ. Observ. Astrophys. Central Russia*, v. 2, 1933, p. 188.**[10]**D. W. Peaceman and H. H. Rachford Jr.,*The numerical solution of parabolic and elliptic differential equations*, J. Soc. Indust. Appl. Math.**3**(1955), 28–41. MR**0071874****[11]**Robert D. Richtmyer,*Difference methods for initial-value problems*, Interscience tracts in pure and applied mathematics. Iract 4, Interscience Publishers, Inc., New. York, 1957. MR**0093918****[12]**Jim Douglas Jr.,*The application of stability analysis in the numerical solution of quasi-linear parabolic differential equations*, Trans. Amer. Math. Soc.**89**(1958), 484–518. MR**0131673**, https://doi.org/10.1090/S0002-9947-1958-0131673-9**[13]**Milton Lees,*A priori estimates for the solutions of difference approximations to parabolic partial differential equations*, Duke Math. J.**27**(1960), 297–311. MR**0121998****[14]**Richard S. Varga,*Matrix iterative analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR**0158502****[15]**G. Fairweather & A. R. Mitchell, "Some computational results of an improved ADI method for the Dirichlet problem,"*Comput. J.*(To appear.)

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DOI:
https://doi.org/10.1090/S0025-5718-1967-0221785-2

Article copyright:
© Copyright 1967
American Mathematical Society